II The Moduli Space of Curves Definition, Coordinatization, and Some Properties

In the previous lecture, we studied each curve separately. We now want in its own right the space of all curves of genus g, which we denote by very important is the allied space  

Let us begin as before by looking first at the simplest cases  

II) 97ti,o = SDti.i = A] (the affine line17 with coordinate j). 

Because curves of genus 1 are groups, their automorphisms act transitively on them, hence Wlg,o = 9Jl3)i. To determine this space, recall that all such curves are isomorphic to one of the plane cubics C, defined by 

Out of habit, I find it more comfortable to call affine n-space An instead of C1 because An also denotes affine n-space over other ground fields. 

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Lecture II The Moduli Space of Curves Definition, Coordinatization, and Some Properties... 

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